Szegedy's quantum walk quadratically accelerates reversible classical Markov chains, but extending this mechanism to efficiently implementable quantum Gibbs samplers has remained challenging beyond special classes. We present a walk-free quantum algorithm for preparing purified Gibbs states with a quadratic improvement in spectral-gap dependence for a broad class of quantum Gibbs samplers that satisfy exact Kubo-Martin-Schwinger detailed balance. Our main structural result is an explicit factorization of the corresponding parent Hamiltonian into noncommutative first-order operators. This turns purified Gibbs-state preparation into a singular-value filtering problem and enables a quantum singular value transformation algorithm with quadratically improved gap dependence under standard coherent-access and warm-start assumptions. The framework applies to several efficiently implementable Gibbs samplers beyond the Davies setting. Moreover, we construct noncommuting Hamiltonians with efficiently preparable warm starts whose refinement under quantum Gibbs samplers requires exponential time. For these models, our algorithm achieves an end-to-end quantum speedup over the worst-case mixing scale.Szegedy's quantum walk quadratically accelerates reversible classical Markov chains, but extending this mechanism to efficiently implementable quantum Gibbs samplers has remained challenging beyond special classes. We present a walk-free quantum algorithm for preparing purified Gibbs states with a quadratic improvement in spectral-gap dependence for a broad class of quantum Gibbs samplers that satisfy exact Kubo-Martin-Schwinger detailed balance. Our main structural result is an explicit factorization of the corresponding parent Hamiltonian into noncommutative first-order operators. This turns purified Gibbs-state preparation into a singular-value filtering problem and enables a quantum singular value transformation algorithm with quadratically improved gap dependence under standard coherent-access and warm-start assumptions. The framework applies to several efficiently implementable Gibbs samplers beyond the Davies setting. Moreover, we construct noncommuting Hamiltonians with efficiently preparable warm starts whose refinement under quantum Gibbs samplers requires exponential time. For these models, our algorithm achieves an end-to-end quantum speedup over the worst-case mixing scale.

quant-pharXiv:2604.22996Apr 28, 20260 citations

Gibbs state preparation for commuting Hamiltonian: Mapping to classical Gibbs sampling

Yeongwoo Hwang, Jiaqing Jiang

Gibbs state preparation, or Gibbs sampling, is a key computational technique extensively used in physics, statistics, and other scientific fields. Recent efforts for designing fast mixing Gibbs samplers for quantum Hamiltonians have largely focused on commuting local Hamiltonians (CLHs), a non-trivial subclass of Hamiltonians which include highly entangled systems such as the Toric code and quantum double model. Most previous Gibbs samplers relied on simulating the Davies generator, which is a Lindbladian associated with the thermalization process in nature.Instead of using the Davies generator, we design a different Gibbs sampler for various CLHs by giving a reduction to classical Hamiltonians, in the sense that one can efficiently prepare the Gibbs state for some CLH H on a quantum computer as long as one can efficiently do classical Gibbs sampling for the corresponding classical Hamiltonian H ( c ) . We demonstrate that our Gibbs sampler is able to replicate state-of-the-art results as well as prepare the Gibbs state in regimes which were previously unknown, such as the low temperature region, as long as there exists fast mixing Gibbs samplers for the corresponding classical Hamiltonians. Our reductions are as follows.– If H is a 2-local qudit CLH, then H ( c ) is a 2-local qudit classical Hamiltonian.– If H is a 4-local qubit CLH on 2D lattice and there are no classical qubits, then H ( c ) is a 2-local qudit classical Hamiltonian on a planar graph. As an example, our algorithm can prepare the Gibbs state for the (defected) Toric code at any non-zero temperature in O ( n 2 p o l y ( l o g n ) ) time.– If H is a 4-local qubit CLH on 2D lattice and there are classical qubits, assuming that quantum terms are uniformly correctable, then H ( c ) is a constant-local classical Hamiltonian.Gibbs state preparation, or Gibbs sampling, is a key computational technique extensively used in physics, statistics, and other scientific fields. Recent efforts for designing fast mixing Gibbs samplers for quantum Hamiltonians have largely focused on commuting local Hamiltonians (CLHs), a non-trivial subclass of Hamiltonians which include highly entangled systems such as the Toric code and quantum double model. Most previous Gibbs samplers relied on simulating the Davies generator, which is a Lindbladian associated with the thermalization process in nature.Instead of using the Davies generator, we design a different Gibbs sampler for various CLHs by giving a reduction to classical Hamiltonians, in the sense that one can efficiently prepare the Gibbs state for some CLH H on a quantum computer as long as one can efficiently do classical Gibbs sampling for the corresponding classical Hamiltonian H ( c ) . We demonstrate that our Gibbs sampler is able to replicate state-of-the-art results as well as prepare the Gibbs state in regimes which were previously unknown, such as the low temperature region, as long as there exists fast mixing Gibbs samplers for the corresponding classical Hamiltonians. Our reductions are as follows.– If H is a 2-local qudit CLH, then H ( c ) is a 2-local qudit classical Hamiltonian.– If H is a 4-local qubit CLH on 2D lattice and there are no classical qubits, then H ( c ) is a 2-local qudit classical Hamiltonian on a planar graph. As an example, our algorithm can prepare the Gibbs state for the (defected) Toric code at any non-zero temperature in O ( n 2 p o l y ( l o g n ) ) time.– If H is a 4-local qubit CLH on 2D lattice and there are classical qubits, assuming that quantum terms are uniformly correctable, then H ( c ) is a constant-local classical Hamiltonian.

arXiv:2410.04909QuantumSep 18, 20261 citation

Modified logarithmic Sobolev inequality for 1D non-commuting Hamiltonians

Ángela Capel, David Pérez-García, Matteo Scandi

Given a finite-range, non-commuting Hamiltonian on a 1D chain of finite spins, we prove a modified logarithmic Sobolev inequality with constant $Ω(1/\log n)$ for the heat-bath dynamics, a quasi-local Gibbs sampler, and for the regularised heat-bath dynamics. This implies rapid mixing of both samplers towards the Gibbs state in relative entropy and trace norm, with mixing times of order $\mathcal{O}(\log n\cdot \log(n/ε))$ . The interactions may be spatially inhomogeneous. The proof combines weak quasi-factorisation of relative entropy, decay-of-correlation estimates, a uniform conditional local gap, and a buffered comparison between conditional entropy and quadratic energy. The resulting bounded entropy defect is removed using the global gap. To the best of our knowledge, this is the first proof of MLSI in the non-commuting regime.Given a finite-range, non-commuting Hamiltonian on a 1D chain of finite spins, we prove a modified logarithmic Sobolev inequality with constant $Ω(1/\log n)$ for the heat-bath dynamics, a quasi-local Gibbs sampler, and for the regularised heat-bath dynamics. This implies rapid mixing of both samplers towards the Gibbs state in relative entropy and trace norm, with mixing times of order $\mathcal{O}(\log n\cdot \log(n/ε))$. The interactions may be spatially inhomogeneous. The proof combines weak quasi-factorisation of relative entropy, decay-of-correlation estimates, a uniform conditional local gap, and a buffered comparison between conditional entropy and quadratic energy. The resulting bounded entropy defect is removed using the global gap. To the best of our knowledge, this is the first proof of MLSI in the non-commuting regime.

quant-pharXiv:2610.03683Oct 5, 20260 citations

Gibbs-like ensembles and the inference of pure states

M. Casas, Angel Ricardo Plastino

Physica A Statistical Mechanics and its ApplicationsJul 1, 19974 citations

Given a finite-range, non-commuting Hamiltonian on a 1D chain of finite spins, we prove a modified logarithmic Sobolev inequality with constant

Ω(1/\log n)

for the heat-bath dynamics, a quasi-local Gibbs sampler, and for the regularised heat-bath dynamics. This implies rapid mixing of both samplers towards the Gibbs state in relative entropy and trace norm, with mixing times of order

\mathcal{O}(\log n\cdot \log(n/ε))

. The interactions may be spatially inhomogeneous. The proof combines weak quasi-factorisation of relative entropy, decay-of-correlation estimates, a uniform conditional local gap, and a buffered comparison between conditional entropy and quadratic energy. The resulting bounded entropy defect is removed using the global gap. To the best of our knowledge, this is the first proof of MLSI in the non-commuting regime.